Vinogradov, who was born at Milolyub (now Velikiye Luki) in Russia, held a number of posts at various institutions in the Soviet Union. Initially he taught at the University of Perm (1918–20) until appointed professor of mathematics at the Leningrad Polytechnic Institute. In 1925 he became a professor at Leningrad State University and in 1932 was appointed chairman of the National Committee of Soviet Mathematicians of the Soviet Academy of Sciences. From 1934 he was professor of mathematics at Moscow State University.

Vinogradov has been preeminent in the field of analytical number theory, i.e., the study of problems posed in purely number-theoretic terms by means of the techniques of analysis. He published several books, chiefly on various aspects of number theory, which include A New Method in the Analytical Theory of Numbers (1937) and The Method of Trigonometric Sums in the Theory of Numbers (1947).

One of his most outstanding achievements was his solution in 1937 of a problem of Christian Goldbach's (not to be confused with Goldbach's famous conjecture about even numbers and primes, which is still undecided). Vinogradov showed that every sufficiently large odd natural number is a sum of three primes.